Understanding Kinetic and Potential Energy in Physics

Potential Energy: Stored Energy

Potential energy ($E_p$ or $PE$) is "stored" energy. It is energy that an object has because of its position relative to other objects or because of its internal configuration. There are two primary types we study in introductory physics: gravitational and elastic.

Gravitational Potential Energy

When an object is lifted against the force of gravity, work is done to move it. This work is stored as gravitational potential energy ($GPE$). The amount of energy depends on how heavy the object is, how fast gravity is pulling on it, and how high it is from a reference point.

The formula for gravitational potential energy is:

$$E_p = m g h$$

Where:

• $m$ is the mass ($\text{kg}$)
• $g$ is the acceleration due to gravity (approximately $9.8\ \text{m/s}^2$ on Earth)
• $h$ is the height above a chosen reference point ($\text{m}$)

Elastic Potential Energy

Not all stored energy comes from height. Some energy is stored by deforming an object, such as stretching a rubber band or compressing a spring. This is known as elastic potential energy.

While the formula for elastic energy can become more complex depending on the material, it fundamentally relies on the force applied and the distance the object is stretched or compressed.

The Relationship: Energy Transformation

One of the most important principles in physics is that energy is not created or destroyed; it only changes form. This is known as the Law of Conservation of Energy. In a closed system without friction, the total mechanical energy (the sum of kinetic and potential energy) remains constant.

$$E_{\text{total}} = E_k + E_p = \text{constant}$$

Consider a roller coaster car at the top of a hill. At its highest point, its velocity is zero, meaning its kinetic energy is zero, but its gravitational potential energy is at its maximum. As the car plunges down the hill, gravity pulls it downward, converting that stored potential energy into kinetic energy. As the car speeds up, $E_p$ decreases while $E_k$ increases.

Worked Example: The Falling Object

To see how these formulas work in practice, let's solve a common problem.

Problem: A $2\ \text{kg}$ book is held at a height of $5\ \text{meters}$ above the floor. Calculate its gravitational potential energy. Then, calculate its kinetic energy at the exact moment before it hits the floor (assume no air resistance).

Step 1: Calculate Potential Energy ($E_p$) Using the formula $E_p = mgh$:

$$\begin{aligned} E_p &= 2\ \text{kg} \cdot 9.8\ \text{m/s}^2 \cdot 5\ \text{m} \\ E_p &= 98\ \text{J} \end{aligned}$$

Step 2: Calculate Kinetic Energy ($E_k$) According to the Law of Conservation of Energy, if we ignore air resistance, all the potential energy at the top will convert into kinetic energy at the bottom.

$$\begin{aligned} E_k &= E_p \\ E_k &= 98\ \text{J} \end{aligned}$$

By knowing the kinetic energy at the bottom, you could even work backward to find the velocity of the book using $v = \sqrt{\frac{2E_k}{m}}$.